Fixed Points of the Smoothing Transform: the Boundary Case

Let A=(A₁,A₂,A₃,…) be a random sequence of non-negative numbers that are ultimately zero with E[∑ Aᵢ]=1 and E [∑ Aᵢ log Aᵢ ] ≤ 0. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation Φ(ψ)= E [ ∏ᵢ Φ(ψ Aᵢ) ], where Φ is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when E[∑ Aᵢ log Aᵢ ]<0. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where E[∑ Aᵢ log Aᵢ ]=0, are obtained.